"Puzzle" is such a huge genre. If you're looking for "block falling/moving" type stuff, I'd rank the Puzzle League series near the top. Tetris Attack (Panel de Pon) and Pokemon Puzzle League, specifically. Great easy mechanics, with fair steadily-increasing challenge. There are some more recent handheld entries as well.

The Magical Drop games are also a lot of fun, though beware -- they originated in the arcade and it's thus a very hard quarter-munching type of series.

Now as for puzzle-platformers and games with playable characters.... Lemmings and The Lost Vikings are unstoppable.